Asymptotes of a²b²+2ab²x+... curve

Curve: a²b² + 2ab²x + b²x² + a²x² + 2ax³ + x⁴ − x²y² = 0.

Group the terms: b²(a² + 2ax + x²) + x²(a² + 2ax + x²) = (a + x)²(b² + x²). So the curve is

(a + x)²(b² + x²) = x²y², degree 4.

Parallel to the y-axis: coefficient of y² is −x²; x² = 0 gives x = 0.

Oblique:

φ₄(m) = −m² + 1 = 1 − m²
φ₃(m) = 2a

Slopes m = ±1. c = −φ₃/φ₄′ = −2a/(−2m) = a/m.

• m = 1: c = a, so y = x + a.
• m = −1: c = −a, so y = −x − a.

Answer: x = 0, y = x + a, y = −(x + a) (same as the curve (a+x)²(b²+x²) = x²y²).